History
For u = 0 above, the statement is also known as Bolzano's theorem. This theorem was first proved by Bernard Bolzano in 1817. Cauchy provided a proof in 1821. Both were inspired by the goal of formalizing the analysis of functions and the work of Lagrange. The idea that continuous functions possess the intermediate value property has an earlier origin. Simon Stevin proved the intermediate value theorem for polynomials (using a cubic as an example) by providing an algorithm for constructing the decimal expansion of the solution. The algorithm iteratively subdivides the interval into 10 parts, producing an additional decimal digit at each step of the iteration. Before the formal definition of continuity was given, the intermediate value property was given as part of the definition of a continuous function. Proponents include Louis Arbogast, who assumed the functions to have no jumps, satisfy the intermediate value property and have increments whose sizes corresponded to the sizes of the increments of the variable. Earlier authors held the result to be intuitively obvious, and requiring no proof. The insight of Bolzano and Cauchy was to define a general notion of continuity (in terms of infinitesimals in Cauchy's case, and using real inequalities in Bolzano's case), and to provide a proof based on such definitions.
Read more about this topic: Intermediate Value Theorem
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